Problem: The equation of hyperbola $H$ is $\dfrac {(y+6)^{2}}{4}-\dfrac {(x+8)^{2}}{9} = 1$. What are the asymptotes?
We want to rewrite the equation in terms of $y$ , so start off by moving the $y$ terms to one side: $\dfrac {(y+6)^{2}}{4} = 1 + \dfrac {(x+8)^{2}}{9}$ Multiply both sides of the equation by $4$ $(y+6)^{2} = { 4 + \dfrac{ (x+8)^{2} \cdot 4 }{9}}$ Take the square root of both sides. $\sqrt{(y+6)^{2}} = \pm \sqrt { 4 + \dfrac{ (x+8)^{2} \cdot 4 }{9}}$ $ y + 6 = \pm \sqrt { 4 + \dfrac{ (x+8)^{2} \cdot 4 }{9}}$ As $x$ approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it. $y + 6 \approx \pm \sqrt {\dfrac{ (x+8)^{2} \cdot 4 }{9}}$ $y + 6 \approx \pm \left(\dfrac{2 \cdot (x + 8)}{3}\right)$ Subtract $6$ from both sides and rewrite as an equality in terms of $y$ to get the equation of the asymptotes: $y = \pm \dfrac{2}{3}(x + 8) -6$